The standard error (SE) of a statistic (usually an estimate of a parameter) is the standard deviation of its sampling distribution or an estimate of that standard deviation. If the statistic is the sample mean, it is called the standard error of the mean (SEM). The standard error is a key ingredient in producing confidence intervals.
The sampling distribution of a mean is generated by repeated sampling from the same population and recording of the sample means obtained. This forms a distribution of different means, and this distribution has its own mean and variance. Mathematically, the variance of the sampling mean distribution obtained is equal to the variance of the population divided by the sample size. This is because as the sample size increases, sample means cluster more closely around the population mean.
Therefore, the relationship between the standard error of the mean and the standard deviation is such that, for a given sample size, the standard error of the mean equals the standard deviation divided by the square root of the sample size. In other words, the standard error of the mean is a measure of the dispersion of sample means around the population mean.
In regression analysis, the term "standard error" refers either to the square root of the reduced chi-squared statistic or the standard error for a particular regression coefficient (as used in, say, confidence intervals).
Standard error of the sample mean
Exact value
Suppose a statistically independent sample of observations is taken from a statistical population with a standard deviation of . The mean value calculated from the sample, , will have an associated standard error on the mean, , given by:
Practically this tells us that when trying to estimate the value of a population mean, due to the factor , reducing the error on the estimate by a factor of two requires acquiring four times as many observations in the sample; reducing it by a factor of ten requires a hundred times as many observations.
Estimate
The standard deviation of the population being sampled is seldom known. Therefore, the standard error of the mean is usually estimated by replacing with the sample standard deviation instead:
As this is only an estimator for the true "standard error", it is common to see other notations here such as:
A common source of confusion occurs when failing to distinguish clearly between:
the standard deviation of the population (), the standard deviation of the sample (), the standard deviation of the mean itself (, which is the standard error), and the estimator of the standard deviation of the mean (, which is the most often calculated quantity, and is also often colloquially called the standard error).
Accuracy of the estimator
When the sample size is small, using the standard deviation of the sample instead of the true standard deviation of the population will tend to systematically underestimate the population standard deviation, and therefore also the standard error. With n = 2, the underestimate is about 25%, but for n = 6, the underestimate is only 5%. Gurland and Tripathi (1971) provide a correction and equation for this effect. Sokal and Rohlf (1981) give an equation of the correction factor for small samples of n < 20. See unbiased estimation of standard deviation for further discussion.
Derivation
The standard error on the mean may be derived from the variance of a sum of independent random variables, given the definition of variance and some simple properties thereof. If is a sample of independent observations from a population with mean and standard deviation , then we can define the total
which due to the Bienaymé formula, will have variance
where we've approximated the standard deviations, i.e., the uncertainties, of the measurements themselves with the best value for the standard deviation of the population. The mean of these measurements is simply given by
The variance of the mean is then
The standard error is, by definition, the standard deviation of which is simply the square root of the variance:
For correlated random variables the sample variance needs to be computed according to the Markov chain central limit theorem.
Independent and identically distributed random variables with random sample size
There are cases when a sample is taken without knowing, in advance, how many observations will be acceptable according to some criterion. In such cases, the sample size is a random variable whose variation adds to the variation of such that, which follows from the law of total variance.